Subgroups — #15762712

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718354	C₂	1	1	yes	maximal, characteristic
2	#13115797	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115797	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115642	C₂	1	1	yes	maximal, characteristic
5	#13115642	C₂	1	1	yes	maximal, characteristic
6	#13114964	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6095	—	2	2	no	—
9	C₂×#88128	C₂²	1	1	yes	characteristic
10	C₂×#88123	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88061	—	2	2	no	—
15	C₂×#88061	—	2	2	no	—
16	#11718596	C₂²	1	1	yes	characteristic
17	#11718596	C₂²	1	1	yes	characteristic
18	#11718354	—	2	2	no	—
19	#11718354	C₄	1	4	yes	complemented
20	C₁₃×#322	—	2	2	no	—
21	#11718143	—	2	2	no	—
22	#11718143	—	2	2	no	—
23	#11718143	—	2	2	no	—
24	C₂²×D₅₂	—	2	1	no	—
25	C₂²×D₅₂	—	2	1	no	—
26	C₂²×D₅₂	—	2	1	no	—
27	C₂²×Dic₂₆	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	C₂×C₄⋊Dic₁₃	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃	—	2	1	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×#6095	—	2	8	no	—
41	C₂×#6095	—	4	4	no	—
42	C₂×#6095	—	4	4	no	—
43	#88128	—	2	1	no	—
44	#88128	—	2	1	no	—
45	#88128	—	2	1	no	—
46	#88128	C₂×C₄	1	2	yes	—
47	#88123	—	2	1	no	—
48	#88123	—	2	1	no	—
49	#88123	—	2	1	no	—
50	#88123	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₁₃	D₄	1	2	yes	—
57	#88061	—	4	4	no	—
58	#88061	—	4	4	no	—
59	#9829	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	4	1	no	—
65	C₂²×Dic₁₃	—	4	1	no	—
66	C₂²×Dic₁₃	—	4	1	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	2	2	no	—
70	C₂×D₅₂	—	4	2	no	—
71	C₂×D₅₂	—	4	2	no	—
72	C₂×D₅₂	—	4	2	no	—
73	C₂×D₅₂	—	4	2	no	—
74	C₂×D₅₂	—	4	2	no	—
75	C₂×D₅₂	—	4	2	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	1	no	—
78	C₂×D₅₂	—	4	1	no	—
79	C₂×Dic₂₆	—	2	2	no	—
80	C₂×Dic₂₆	—	4	2	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
91	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃	#41	1	1	yes	characteristic
95	C₄²×C₁₃	#41	1	1	yes	characteristic
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	2	no	—
101	C₄⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	—	4	2	no	—
103	C₄⋊Dic₁₃	—	4	2	no	—
104	C₄⋊Dic₁₃	—	4	2	no	—
105	C₄⋊Dic₁₃	—	4	2	no	—
106	#6095	—	4	16	no	—
107	C₂×#492	—	13	1	no	—
108	#1990	—	13	1	no	—
109	#1990	—	13	1	no	—
110	#1957	—	13	1	no	—
111	#1957	—	13	1	no	—
112	#1834	—	13	1	no	—
113	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	#127	1	1	yes	characteristic
115	C₂²×D₁₃	—	4	2	no	—
116	C₂²×D₁₃	—	8	1	no	—
117	C₂²×D₁₃	—	8	1	no	—
118	C₂²×D₁₃	—	8	1	no	—
119	C₂²×D₁₃	—	8	1	no	—
120	C₂²×D₁₃	—	8	1	no	—
121	C₂²×D₁₃	—	8	1	no	—
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₁₃	#135	1	2	yes	—
130	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₃	#137	1	2	yes	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	8	1	no	—
136	C₂×Dic₁₃	—	8	1	no	—
137	C₂×Dic₁₃	—	8	1	no	—
138	C₂×Dic₁₃	—	8	1	no	—
139	C₂×Dic₁₃	—	8	1	no	—
140	C₂×Dic₁₃	—	8	1	no	—
141	C₂×Dic₁₃	—	8	1	no	—
142	C₂×Dic₁₃	—	8	1	no	—
143	D₅₂	—	4	4	no	—
144	D₅₂	—	4	4	no	—
145	D₅₂	—	4	4	no	—
146	D₅₂	—	8	2	no	—
147	D₅₂	—	8	2	no	—
148	D₅₂	—	8	2	no	—
149	Dic₂₆	—	4	4	no	—
150	Dic₂₆	—	8	2	no	—
151	C₈×C₁₃	—	2	4	no	—
152	C₈×C₁₃	—	2	4	no	—
153	C₈×C₁₃	—	4	2	no	—
154	C₂²×SD₁₆	—	26	2	no	—
155	C₂×C₄⋊Q₈	—	13	1	no	—
156	C₂×#127	—	13	1	no	—
157	#492	—	13	4	no	—
158	#492	—	26	2	no	—
159	C₂×#102	—	26	2	no	—
160	C₂×#102	—	26	2	no	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
164	#384	—	13	1	no	—
165	#384	—	13	1	no	—
166	#340	—	26	2	no	—
167	#340	—	26	2	no	—
168	#340	—	26	2	no	—
169	#322	—	2	2	no	—
170	C₂²×C₁₃	#390	1	1	yes	characteristic
171	C₂²×C₁₃	#492	1	1	yes	characteristic
172	C₂²×C₁₃	#495	1	1	yes	characteristic
173	C₂²×C₁₃	#496	1	1	yes	characteristic
174	C₂²×C₁₃	#496	1	1	yes	characteristic
175	C₂²×C₁₃	#496	1	1	yes	characteristic
176	C₂²×C₁₃	#497	1	1	yes	characteristic
177	C₂×D₁₃	—	8	2	no	—
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	16	1	no	—
181	C₂×D₁₃	—	16	1	no	—
182	C₂×D₁₃	—	16	1	no	—
183	C₂×D₁₃	—	16	1	no	—
184	C₄×C₁₃	#418	1	2	yes	—
185	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
186	C₄×C₁₃	#440	1	2	yes	—
187	C₄×C₁₃	#465	1	2	yes	—
188	C₄×C₁₃	#468	1	2	yes	—
189	C₄×C₁₃	#490	1	2	yes	—
190	Dic₁₃	—	8	2	no	—
191	Dic₁₃	—	16	1	no	—
192	Dic₁₃	—	16	1	no	—
193	C₂²×Q₈	—	26	1	no	—
194	C₂²×D₄	—	26	1	no	—
195	C₂²×D₄	—	26	1	no	—
196	C₂²×D₄	—	26	1	no	—
197	C₂×SD₁₆	—	26	8	no	—
198	C₂×SD₁₆	—	52	4	no	—
199	C₂×SD₁₆	—	52	4	no	—
200	C₂²×C₈	D₅₂	1	2	yes	—
201	C₂²×C₈	D₅₂	1	2	yes	—
202	C₂²×C₈	D₅₂	1	2	yes	—
203	C₄⋊Q₈	—	13	2	no	—
204	C₄⋊Q₈	—	26	1	no	—
205	C₄⋊Q₈	—	26	1	no	—
206	C₄⋊Q₈	—	26	1	no	—
207	#127	—	13	2	no	—
208	#127	—	26	1	no	—
209	#127	—	26	1	no	—
210	#127	—	26	1	no	—
211	C₂×C₄⋊C₄	—	26	1	no	—
212	C₂×C₄⋊C₄	—	26	1	no	—
213	C₂×C₄⋊C₄	—	26	1	no	—
214	C₂×C₄⋊C₄	—	26	1	no	—
215	C₂×C₄⋊C₄	—	26	1	no	—
216	C₂×C₄⋊C₄	—	26	1	no	—
217	C₂×C₄⋊C₄	—	26	1	no	—
218	C₂×C₄⋊C₄	—	26	1	no	—
219	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
220	#102	—	52	4	no	—
221	#102	—	52	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	C₄×D₁₃	1	2	yes	—
227	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
228	C₂×C₁₃	#1714	1	1	yes	characteristic
229	C₂×C₁₃	#1715	1	1	yes	characteristic
230	C₂×C₁₃	#1717	1	1	yes	characteristic
231	C₂×C₁₃	#1719	1	1	yes	characteristic
232	C₂×C₁₃	#1961	1	1	yes	characteristic
233	C₂×C₁₃	#1968	1	1	yes	characteristic
234	C₂×C₁₃	#1973	1	1	yes	characteristic
235	D₁₃	—	16	2	no	—
236	C₂⁴	—	52	1	no	—
237	C₂×Q₈	—	26	2	no	—
238	C₂×Q₈	—	52	1	no	—
239	C₂×Q₈	—	52	2	no	—
240	C₂×Q₈	—	52	2	no	—
241	C₂×D₄	—	26	2	no	—
242	C₂×D₄	—	26	2	no	—
243	C₂×D₄	—	26	2	no	—
244	C₂×D₄	—	52	1	no	—
245	C₂×D₄	—	52	2	no	—
246	C₂×D₄	—	52	2	no	—
247	C₂×D₄	—	52	2	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	2	no	—
250	C₂×D₄	—	52	2	no	—
251	C₂×D₄	—	52	1	no	—
252	C₂×D₄	—	52	1	no	—
253	C₂²×C₄	—	52	1	no	—
254	C₂²×C₄	—	52	1	no	—
255	C₂²×C₄	—	52	1	no	—
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
259	SD₁₆	—	52	16	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	#6095	1	4	yes	complemented
268	C₂×C₈	#6103	1	4	yes	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄⋊C₄	—	52	1	no	—
272	C₄⋊C₄	—	52	1	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₄⋊C₄	—	52	2	no	—
275	C₄⋊C₄	—	52	2	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
280	C₄²	#6127	1	1	yes	characteristic
281	C₄²	#6127	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₁₃	#9829	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	52	2	no	—
285	C₂³	—	104	1	no	—
286	C₂³	—	104	1	no	—
287	C₂³	—	104	1	no	—
288	C₂³	—	104	1	no	—
289	C₂³	—	104	1	no	—
290	C₂³	—	104	1	no	—
291	C₂³	#88128	1	1	yes	characteristic
292	Q₈	—	52	4	no	—
293	Q₈	—	104	2	no	—
294	D₄	—	52	4	no	—
295	D₄	—	52	4	no	—
296	D₄	—	52	4	no	—
297	D₄	—	104	2	no	—
298	D₄	—	104	2	no	—
299	D₄	—	104	2	no	—
300	C₂×C₄	—	52	2	no	—
301	C₂×C₄	—	104	1	no	—
302	C₂×C₄	—	104	1	no	—
303	C₂×C₄	—	104	1	no	—
304	C₂×C₄	—	104	1	no	—
305	C₂×C₄	—	104	1	no	—
306	C₂×C₄	—	104	1	no	—
307	C₂×C₄	—	104	1	no	—
308	C₂×C₄	—	104	1	no	—
309	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
310	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
311	C₂×C₄	#88129	1	1	yes	characteristic
312	C₂×C₄	C₂×#6095	1	2	yes	—
313	C₂×C₄	#88158	1	2	yes	—
314	C₂×C₄	#88162	1	2	yes	—
315	C₂×C₄	#88162	1	2	yes	—
316	C₂×C₄	#88162	1	2	yes	—
317	C₂×C₄	#88163	1	2	yes	—
318	C₂×C₄	C₂×#6103	1	1	yes	characteristic
319	C₂×C₄	#88184	1	1	yes	characteristic
320	C₂×C₄	#88184	1	1	yes	characteristic
321	C₈	—	2	4	no	—
322	C₈	—	2	4	no	—
323	C₈	—	4	2	no	—
324	C₂²	—	104	2	no	—
325	C₂²	—	104	2	no	—
326	C₂²	—	104	2	no	—
327	C₂²	—	208	1	no	—
328	C₂²	—	208	1	no	—
329	C₂²	—	208	1	no	—
330	C₂²	—	208	1	no	—
331	C₂²	#11718354	1	1	yes	characteristic
332	C₂²	#11718357	1	1	yes	characteristic
333	C₂²	#11718373	1	1	yes	characteristic
334	C₂²	#11718373	1	1	yes	characteristic
335	C₂²	#11718373	1	1	yes	characteristic
336	C₂²	#11718376	1	1	yes	characteristic
337	C₂²	#11718600	1	1	yes	characteristic
338	C₄	—	104	2	no	—
339	C₄	—	208	1	no	—
340	C₄	—	208	1	no	—
341	C₄	C₄×#6095	1	2	yes	—
342	C₄	#11718371	1	2	yes	—
343	C₄	#11718374	1	2	yes	—
344	C₄	#11718775	1	2	yes	—
345	C₄	#11718792	1	2	yes	—
346	C₄	#11718795	1	2	yes	—
347	C₂	—	208	2	no	—
348	C₂	#13114458	1	1	yes	characteristic
349	C₂	#13114464	1	1	yes	characteristic
350	C₂	#13114475	1	1	yes	characteristic
351	C₂	#13114489	1	1	yes	characteristic
352	C₂	#13115660	1	1	yes	characteristic
353	C₂	#13115690	1	1	yes	characteristic
354	C₂	#13115717	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#9829
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #13115797	C₂	semidirect
3. #13115797	C₂	semidirect
6. #13114964	C₂	semidirect
7. C₂×C₁₃×#322	C₂	semidirect
19. #11718354	C₄	semidirect
91. C₂×C₈×C₁₃	SD₁₆	semidirect
113. C₂×#322	D₁₃	semidirect
267. C₂×C₈	#6095	semidirect
283. C₁₃	#9829	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11718354	2
#13115797	2
#13115642	2
#13114964	2
C₂×C₁₃×#322	2
#9829	13